User blog:Tetramur/Pentational arrays and beyond - comparisons
My comparison would base mainly on two analyses: 1. Hyp cos-Deedlit11-Ikosarakt1 analysis (BEAF-HDI); 2. Bowers-Saibian-Cookiefonster analysis (BEAF-BSC). There is one issue which Saibian wrote in his article: "according to Bowers' the number of entries for a structure is the result by substituting the prime entry for X. Thus X^^(X+1) --> p^^(p+1). The problem here is simply this: what structures is X^^(X+1) actually composed of, and how do they result in p^^(p+1) entries?" I can't help, but this is a question I ask, too. Pentational arrays So, tetrational arrays are well-defined. Let's start with pentational arrays: So, we see how much triakulus, kungulus and many more googologisms are and how great is gap between the two - we reached \(\Gamma_0\) in the end, and BEAF-HDI will wait much more until it reaches this ordinal! Some examples: 1. Primitive sequence number is about \(\{10,10,2 (X \uparrow\uparrow X) 2\}\) or \(f_{\varepsilon_0+1}(10)\). (That means that our systems greatly exceed the power of primitive sequence system, but we didn't even reach the power of Bashicu's pair sequence yet!) 2. Fish number 5 is about \(\{63,63,2 (X \uparrow\uparrow X) 2\}\) or \(f_{\varepsilon_0+1}(63)\). 3. Marxen.c is upper-bounded with \(\{1000000,1000000,1000000,1000000,1000000 (X \uparrow\uparrow X) 2\}\) or \(f_{\varepsilon_0+\omega^3}(1000000)\). 4. Fish number 6 is about \(\{63,63,2 (X \uparrow\uparrow X \uparrow 2) 2\}\) or \(f_{\zeta_0+1}(63)\). So we covered all Bignum Bakeoff and Fish numbers except Loader's number which is impossible to reach using BEAF, maybe. P.S. I suspected that Hyp cos made one or more serious mistakes in his part of work. But now I realized that two systems will diverge even more. P.P.S. We can include these structures in usual arrays and make complex structures. And ordinal of beginning is added to ordinal of structure, when the structure is included into array, and the ordinals of the structures multiply by each other, when we have two or more structures in one array simultaneously. Structures That is a piece of info I got directly from Bowers. Here are some of the structures encountered between X^^X and X^^(X+1): X^^X = X^X^X^X^X......|1 = tetrational space X^^X + 1 = tetration space and one entry X^^X * 2 = 2 tetration spaces X^^X * X = X^(X^^X+1) = a row of tetration spaces X^^X * X^2 = X^(X^^X+2) = a plane of tetration spaces X^(X^^X+X) = an omniverse (X^X) of tetration spaces X^(X^^X+X*2) = an omniverse of omniverse of tetration spaces X^(X^^X+X^2) = a biomniverse of tetration spaces = an omniverse of omniverse of omniverse of omniverse.........of tetration spaces. X^(X^^X+X^3) = a triomniverse of tetration spaces X^(X^^X * 2) = tetration space of tetration spaces X^(X^^X * 3) = tetration space of tetration space of tetration spaces X^(X^^X * X) = X^X^(X^^X+1) = tetration space of tetration space of tetration space of tetration space .............infinity, note that the +1 has only climbed to height 2 on the power tower (superior tetrational space). (X^^X)^(X+1) = X^(X^^X * (X+1)) = tetration space of superior tetrational spaces (X^^X)^(X+2) = X^(X^^X * (X+2)) = tetration space of tetration space of superior tetrational spaces (X^^X)^(X*2) = X^(X^^X * 2X) = X^X^(X^^X+2) = bisuperior tetrational space (X^^X)^(X*3) = X^X^(X^^X+3) - trisuperior tetrational space (X^^X)^(X^2) = (X^^X)^(X*X) = X^X^(X^^X+X) - omnisuperior tetrational space Now for some bigger jumps and gaps: (X^^X)^(X^X) = X^((X^^X)^X) = X^X^(X^^X*X) = X^X^X^(X^^X+1) - note that +1 is on height 3. X^X^X^X^(X^^X+1) - note that +1 is on height 4. As the +1 climbs from height 0 (bottom of power tower) to height X (just above the infinity barrier) we will get to this: X^X^X^X^X...........|1+1 = X^X^X^X.......|2 X^X^X^X^X..........|X = X^^(X+1) Continuing to climb and extending power tower: X^X^X^X^X..........|X^X = X^^(X+2) X^X^X^X^X..........|X^X^X = X^^(X+3) X^X^X^X^X..........|X^X^X^X^X..........|1 = X^^2X X^X^X^X^X..........|X^X^X^X^X..........|X^X^X^X^X..........|1 = X^^3X ... X^X^X^X^X..........|X^X^X^X^X..........|X^X^X^X^X..........|...|1 = X^^X^2 Arrowal arrays Let's go further. We will see "arrowal" arrays, that is, arrays defined using arrows, and "linear array"-arrays. So let's continue comparison: YIKES! We are at SVO now, and this is the point after which BEAF-BSC will be the same ordinal in FGH as BEAF-HDI! But now there is third system - it says that the limit of the legion arrays is LVO, which is much smaller than current limit. I don't know much of it, only the growing limit. So, I will compare two systems: BEAF-BSC (BEAF-HDI is the same after SVO) and weak BEAF. "Weak BEAF" is old BEAF, "middle BEAF" is BEAF-HDI, and "strong BEAF" is BEAF-BSC. Bowers largely agrees with the "strong" BEAF. So, I think that BEAF is fully formalized and well-defined (currently until SVO and maybe even further). But there are many people who call BEAF a waste of time. People mostly abandoned trying to define or make guesses on how powerful BEAF is, and a large portion of members decidedly have their opinion on BEAF that it's meaningless and not worth studying past tetrational arrays. Part 2 of this is here . Category:Blog posts